The least squares problem is an extremely useful device to represent an approximate solution to overdetermined systems, and a QR factorisation is a common method for solving least squares problems. It is often the case that multiple least squares solutions have to be computed with only minor changes in the underlying data. In this case, knowledge of the difference between the old data set and the new one can be used to update an existing QR factorisation at a reduced computational cost. However, fairly recent developments have introduced the widespread use of massively parallel computational devices known as GPUs. GPUs have allowed QR factorisations, and subsequently, least squares solutions to be calculated in a greatly reduced time. The purpose of this project is to investigate the viability of the implementation of QR updating algorithms on the GPU and attempt gain speedup with a GPU based updating algorithm over both existing sequential QR updating algorithms, and full GPU QR factorisations. The conclusion of the investigation is that GPU based updating algorithms gain speedups over their sequential analogues for almost all problem sizes, whereas the proposed algorithms only gain speedups over the full GPU QR factorisation under certain conditions.